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Numerical modelling of dendritic solidification based on phase field formulation and adaptive meshless solution procedure : dissertation
Tadej Dobravec, 2021, doctoral dissertation

Abstract: The main aim of the dissertation is to develop a novel numerical approach for an accurate and computationally efficient modelling of dendritic solidification, which is one of the most commonly observed phenomena in the industrial casting of the metallic alloys. The size and the morphology of dendritic structures as well as the distribution of the solute within them critically effect the mechanical and the electro-chemical properties of the solidified material. The numerical modelling of dendritic solidification can be applied for an in-depth understanding and optimisation of the casting process under various solidification conditions and chemical compositions of the alloy under consideration. The dendritic solidification of pure materials and dilute multi-component alloys is considered in the dissertation. The phase field formulation is applied for the modelling of dendritic solidification. The formulation is based on the introduction of the continuous phase field variable that is constant in the bulk of the solid and liquid phases. The phase field variable has a smooth transition from the value denoting the solid phase to the value denoting the liquid phase at the solid-liquid interface over the characteristic interface thickness. A phase field model yields a system of coupled non-linear parabolic partial differential equations that govern the evolution of the phase field and other thermodynamic variables. The meshless radial basis function-generated finite-differences (RBF-FD) method is used for the spatial discretisation of the system of partial differential equations. The forward Euler scheme is applied for the temporal discretisation. Fifth-degree polyharmonic splines are used as the shape functions in the RBF-FD method. A second-order accurate RBF-FD method is achieved by augmenting the shape functions with monomials up to the second degree. The adaptive solution procedure is developed in order to speed-up the calculations. The procedure is based on the quadtree domain decomposition of a rectangular computational domain into rectangular computational sub-domains of different sizes. Each quadtree sub-domain has its own regular or scattered distribution of computational nodes in which the RBF-FD method and the forward Euler scheme apply for the discretisation of the system of partial differential equations. The adaptive solution procedure dynamically ensures the prescribed highest density of the computational nodes at the solid-liquid interface and the lowest-possible density in the bulk of the solid and liquid phases. The adaptive time-stepping is employed to further speed-up the calculations. The stable time step in the forward Euler scheme depends on the density of the computational nodes; hence, different time steps can be used in quadtree sub-domains with different node densities. The main originality of the present work is the use of the RBF-FD method for the thorough analysis of the impact of the type of the node distribution and the size of a local sub-domain to the accuracy when the phase field modelling of dendritic solidification for arbitrary preferential growth directions is considered. It is shown how the use of the scattered node distribution reduces the undesirable mesh-induced anisotropy effects, present when the partial differential equations are discretisied on a regular node distribution. The main advantage of the RBF-FD method for the phase field modelling of dendritic growth is the simple discretisation of the partial differential equations on the scattered node distributions. The RBF-FD method is, for the first time, used in combination with the spatial-temporal adaptive solution procedure based on the quadtree domain decomposition. The adaptive solution procedure successfully speeds-up the calculations; however, the advantages of the use of the scattered node distribution are partly compromised due to the impact of regularity in the quadtree domain decomposition.
Keywords: dendritic solidification, phase field method, meshless methods, RBF-FD, adaptive solution procedure
Published in RUNG: 07.04.2021; Views: 3162; Downloads: 136
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3.
Modelling of Macrosegregation of a Low-Frequency Electromagnetic Direct Chill Casting by a Meshless Method
Vanja Hatić, 2019, doctoral dissertation

Abstract: The main aim of the dissertation is to develop a meshless model that describes the solidification and macrosegregation phenomena during the direct chill casting (DCC) of aluminium alloys under the influence of a low-frequency electromagnetic field. Macrosegregation is an undesired consequence of alloy solidification. It represents one of the major casting defects and substantially reduces the quality of the finished product. On the other hand, low-frequency electromagnetic casting (LFEC) is a process that promises to increase greatly the product quality, including the reduction of macrosegregation. The modelling of both processes is of tremendous importance to the metallurgical industry, due to the high costs of experiments during production. The volume-averaging formulation is used for the modelling of the solid-liquid interaction. The conservation equations for mass, energy, momentum, and species are used to model the solidification of aluminium-alloy billets in axysimmetry. The electromagnetic-induction equation is coupled with the melt flow. It is used to calculate the magnetic vector potential and the Lorentz force. The Lorentz force is time-averaged and included in the momentum-conservation equation, which intensifies the melt flow. The effect of Joule heating is neglected in the energy conservation due to its insignificant contribution. The semi-continuous casting process is modelled with the Eulerian approach. This implies that the global computational domain is fixed in space. The inflow of the liquid melt is assumed at the top boundary and the outflow of the solid metal is assumed at the bottom. It is assumed that the whole mushy area is a rigid porous media, which is modelled with the Darcy law. The Kozeny-Carman relation is used for the permeability definition. The incompressible mass conservation is ensured by the pressure correction, which is performed with the fractional step method. The conservation equations and the induction equation are posed in the cylindrical coordinate system. A linearised eutectic binary phase diagram is used to predict the solute redistribution in the solid and liquid phases. The micro model uses the lever rule to determine the temperature and the liquid fraction field from the transport equations. The partial differential equations are solved with the meshless-diffuse-approximate method (DAM). The DAM uses weighted least squares to determine a locally smooth approximation from a discrete set of data. The second-order polynomials are used as the trial functions, while the Gaussian function is used as the weight function. The method is localised by defining a smooth approximation for each computational node separately. This is performed by associating each node with a unique local neighbourhood, which is used for the minimisation. There are 14 nodes included in the local subdomains for the DCC and LFEC simulations. The stability of the advective term is achieved with a shift of the Gaussian weight in the upwind direction. This approach is called the adaptive upwind weight function and is used in the DAM for the first time. The Explicit-Euler scheme is used for temporal discretisation. The use of a meshless method and the automatic node-arrangement generation makes it possible to investigate the complicated flow structures, which are formed in geometrically complex inflow conditions in a straightforward way. A realistic inflow geometry and mould can therefore be included in the model. The number of computational nodes is increased in the mushy zone and decreased in the solid phase, due to the optimisation of the computational time and memory. The computational node arrangement is automatically adapted with time, as the position of the mushy zone is changed in shape and position.
Keywords: low-frequency electromagnetic casting, direct chill casting, macrosegregation, electromagnetic stirring, aluminium alloys, meshless methods, diffuse-approximate method, multiphysics model, solidification
Published in RUNG: 25.04.2019; Views: 4044; Downloads: 149
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4.
CONTRIBUTION TO DEVELOPMENT OF MESHLESS METHODS FOR FREE AND MOVING BOUNDARY PROBLEMS
NAZIA TALAT, 2018, doctoral dissertation

Keywords: Two-phase flow, free and moving boundaries, computational fluid dynamics, phasefield formulation, 2D problems, axisymmetric problems, diffuse approximate meshless method, Rayleigh-Taylor instability, Boussinesq approximation, variable density and viscosity, flow focusing, dripping, jetting
Published in RUNG: 11.09.2018; Views: 5197; Downloads: 179  (1 vote)
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5.
Simulation Of Gas Focused Liquid Jets
Rizwan Zahoor, 2018, doctoral dissertation

Abstract: The main aim of dissertation is to develop an experimentally verified computational fluid dynamic (CFD) model of micron-sized liquid jet, produced by an injection molded Gas Dynamic Virtual Nozzle (GDVN). In these nozzles, liquid jets are efficiently orientedly transporting mass and momentum. They are produced by intelligently projecting hydrodynamic focusing effect from a high-speed stream of a co-flowing lower density and lower viscosity gas on a stream of liquid from a feeding capillary. Liquid micro-jets are used for delivery of protein crystal samples in a hard X-ray beam in serial femtosecond crystallography experiments. The diffraction patterns of crystals are collected just before their destruction. The samples are hard to crystallize and very precious, so a thorough knowledge of the jet used in delivering them is required. The jet characteristics are analyzed as a function of operating parameters, geometry and material properties. The physical model is described by mixture formulation and Navier-Stokes equations for transient, Newtonian, two-phase, compressible flow. Multiphase flow problem is solved with finite volume method (FVM), where fluid-fluid interface tracking is obtained with volume of fluid (VOF). The implementation of FVM-VOF CFD model is available in open source codes OpenFOAM and Gerris. They are validated by performing a series of standard interface advection and multiphase flow test cases. Both open source codes are compared for their abilities in solving GDVN flow problem. Due to the compressible nature of the focusing gas flow, OpenFOAM was chosen for GDVN simulations, since Gerris has no compressible flow option. Constant effective material properties are used in the phases together with ideal gas density constitutive relation. A mixture model of the two-phase system is solved in axisymmetry. The discretization of the nozzle and chamber system uses approximately 300 000 finite volumes. Mesh independent results are obtained with the finite volumes of the size 0.25 µm in the vicinity of the jet and drops. The simulations are compared with experimental results according to the jet thickness and length for distilled water jet and helium focusing gas, discharging into low-pressure environment of 150 Pa. Reynolds numbers of the liquid and gas are in the range 413-3828 and 17-1222, respectively and Weber number in the range 3-353. A reasonably good agreement with experimental and scaling results is found for the range of nozzle operating parameters never tackled before. Subsequently, a numerical study of effects of nozzle geometry on stability, shape and flow characteristics of micron-sized liquid jets is performed. The jet characteristics are described as a function of (i) capillary-to-orifice distance, (ii) nozzle outlet orifice diameter and (iii) liquid feeding capillary angle. The study is performed for two sets of liquid flow rates while keeping the gas flow rate unchanged. It is observed that for each value of capillary-to-orifice distance and nozzle outlet diameter, there exists a minimum liquid flow rate below which the jet stability cannot be achieved. It is found that the changes in the nozzle outlet diameter have the biggest influence on the jet diameter, length and velocity, while the liquid capillary angle has no observable effect on the jet characteristic. Change in capillary-to-orifice distance does not affect the flow field around micro jet, so the jet stability and shape is found to be affected by the way liquid-gas interacts near meniscus. The same numerical model is used to additionally analyze the jet performance under the influence of Argon, Carbon dioxide and Nitrogen focusing gases. The study shows that the helium gas at the same mass flow rate provides twice the length of the jet compared to other gases. The jet focused with helium is also much thinner, faster and interestingly shows no considerable temperature drop at the nozzle outlet. This work for the first time discuss the computational model of an injection molded micron-sized nozzle and produces valuable information for their design.
Keywords: Microfluidics, gas dynamic virtual nozzle, flow focusing, micro-jet, convective instability, absolute instability, compressible multiphase flows, dripping, spurting, jetting, jet thickness, jet length, computational fluid dynamics, finite volume method, volume of fluid method
Published in RUNG: 27.03.2018; Views: 6548; Downloads: 170
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6.
Influence of Gas Dynamic Virtual Nozzle Geometry on Micro-Jet Characteristics
Rizwan Zahoor, Saša Bajt, Božidar Šarler, 2018, original scientific article

Abstract: In this paper we present a numerical study investigating the effects of nozzle geometry on stability, shape and flow characteristics of micron-sized liquid jets, produced by injection molded gas dynamic virtual nozzles (GDVNs) operating in vacuum. The jet characteristics are described as a function of (i) capillary-to-orifice distance, (ii) nozzle outlet orifice diameter, and (iii) liquid feeding capillary angle. An experimentally verified numerical model of GDVN with laminar two-phase Newtonian compressible flow, based on finite volume method and volume of fluid interface tracking, is used to assess the changes. The study is performed for two sets of liquid flow rates while keeping the gas flow rate constant. It is observed that for each value of capillary-to-orifice distance and nozzle outlet diameter there is a minimum liquid flow rate below which the jet is unstable. We find that the nozzle outlet diameter has the biggest influence on the jet diameter, length and velocity, while liquid capillary angle has no observable effect on jet characteristic. Varying capillary-to-orifice distance does not affect the flow field around micro-jet. It is found that the liquid and the gas interaction near the meniscus primarily affect the jet stability and shape
Keywords: Gas dynamic virtual nozzle, Micro-jet, Compressible multiphase flow, Finite volume method, Volume of fluid, Jetting, Dripping
Published in RUNG: 09.03.2018; Views: 5158; Downloads: 0
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7.
Meshless modeling of thermo-mechanics of low-frequency electromagnetic direct chill casting
Boštjan Mavrič, 2017, doctoral dissertation

Abstract: The aim of this dissertation is to devise a meshless model describing the thermomechanical phenomena, which occur during DC casting of aluminium alloys under the influence of electromagnetic stirring. The thermoemchanical phenomena are important, because they can cause several type of defects, which can significantly deteriorate the quality of the resulting billet. The two most important of them are the hot tearing, which causes cracks to appear in the mushy zone, and the porosity, which demonstrates itself as micrometer sized voids in the microstructure of the billet. To calculate the stresses and strains, a computational model of the stationary state of the process, stated in axial symmetry, is formulated. It uses Eulerian formulation by fixing the computational domain to the mold of the casting device allowing the material to move through the computational domain. The stresses are calculated from the stress equilibrium equations. The small strain approximation is used to consider the three contributions to strain. The strain consists of the thermal strain, which is caused by the inhomogeneous thermal profile in the billet, the viscoplastic strain, which is caused by the irreversible deformation because of the large stresses occurring in the billet, and the elastic strain. The spatial discretization of the governing equations is performed by local radial basis function collocation method (LRBFCM) and the temporal discretization is achieved by the method of lines with implicit Euler formula. The method used for spatial discretization uses radial basis functions augmented by monomials to approximate the solution values on localized stencils. This approximation is used to construct the discretization coefficients of the differential operators present in the model. A flexible framework for formulation of multiphysics problems is developed to use the obtained discretization coefficients to construct the temporal discretization of the governing equations. The node arrangement, on which the spatial discretization is performed, was generated by a point-repel algorithm. The performance of the method is tested on several benchmark test cases. The accuracy of the discretization is estimated by comparing the analytic and the numerical solution to several stationary problems in thermomechancis. Of special interest is the performance of the method with respect to the choice of the shape parameter, which determines the spatial scale of the radial basis functions. Besides this, the dependence of the condition number of the interpolation matrix on the shape parameter is studied. The condition number is found fit to replace the condition number as the shape-determining free parameter of the method. The implementation of the solver of time dependent problems is tested on problem of thermoelasticity, which couples the thermal transport with the elastic waves. The results of the problem are compared with the finite element method, showing good agreement of the two methods. The results are also compared with the results obtained by meshless local Petrov-Galerkin method and the proposed local collocation method demonstrated significantly better solution quality in the studied case. The performance of the solver used to solve the system of nonlinear equations given by the viscoplastic constitutive equations is estimated on a quasi zero-dimensional problem. The results are found to match perfectly. Solution of a more complicated problem is obtained with the proposed method and the finite-element method, both methods giving practically the same solution, although some serious limitations of the chosen finite element solver are exposed during the selection of the problem parameters. Finally, the devised method is applied to the problem of DC casting of aluminium alloys. The thermomechanical model relies on a model of heat and mass transfer to obtain the input fields needed in the solver. The required fields are: temperature, pressure, liquid
Keywords: thermomechanics, viscoplasticity, aluminium alloys, direct-chill casting, electromagnetic stirring, hot tearing, porosity, meshless methods, local collocation method, radial basis functions, shape parameter
Published in RUNG: 28.06.2017; Views: 5930; Downloads: 256
.pdf Full text (21,30 MB)

8.
A non-singular method of fundamental solutions for two-dimensional steady-state isotropic thermoelasticity problems
qingguo liu, Božidar Šarler, 2017, original scientific article

Abstract: We consider a boundary meshless numerical solution for two-dimensional linear static thermoelastic problems. The formulation of the problem is based on the approach of Marin and Karageorghis, where the Laplace equation for the temperature field is solved first, followed by a particular solution of the non-homogenous term in the Navier-Lamé system for the displacement, the solution of the homogenous equilibrium equations, and finally the application of the superposition principle. The solution of the problem is based on the method of fundamental solutions (MFS) with source points on the boundary. This is, by complying with the Dirichlet boundary conditions, achieved by the replacement of the concentrated point sources with distributed sources over the disk around the singularity, and for complying with the Neumann boundary conditions by assuming a balance of the heat fluxes and the forces. The derived non-singular MFS is assessed by a comparison with analytical solutions and the MFS for problems that can include different materials in thermal and mechanical contact. The method is easy to code, accurate, efficient and represents a pioneering attempt to solve thermoelastic problems with a MFS-type method without an artificial boundary. The procedure makes it possible to solve a broad spectra of thermomechanical problems.
Keywords: Keywords: Isotropic thermoelasticity Meshless method Non-singular method of fundamental solutions Collocation Efficient desingularisation
Published in RUNG: 23.12.2016; Views: 4557; Downloads: 0
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9.
Method of Regularized Sources for Stokes Flow Problems with Improved Calculation of Velocity Derivatives at the Boundary
Božidar Šarler, wen shiting, li ming, published scientific conference contribution abstract

Abstract: The solution of Stokes flow problems with Dirichlet and Neumann boundary conditions is performed by a non-singular Method of Fundamental Solutions which does not require artificial boundary, i.e. source points of fundamental solution coincide with the collocation points on the boundary. Instead of Dirac delta force, an exponential function, called blob, with a free parameter epsilon is employed, which limits to Dirac delta function when epsilon limits to zero. The solution of the problem is sought as a linear combination of the fields due to the regularized sources that coincide with the boundary and with their intensities chosen in such a way that the solution complies with the boundary conditions. A two-dimensional flow between parallel plates is chosen to assess the properties of the method. The results of the method are accurate except for the derivatives at the boundary. A correction of the method is proposed which can be used to properly assess also the derivatives at the boundary
Keywords: stokes flow, method of regularized sources, meshless method
Published in RUNG: 28.06.2016; Views: 4331; Downloads: 0
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